Minimum-Distortion Wealth Taxation, I: Information-Theoretic versus Transport-Geometric Optimality on the Proportional Class

Technical Summary: Minimum-Distortion Wealth Taxation, I

Problem Statement
This paper investigates the design of minimum-distortion wealth taxation within a Fokker–Planck (FP) framework on log-wealth. It addresses a fundamental normative divide in optimal tax theory: the tension between minimizing allocational distortion (preserving household portfolio decisions) and minimizing distributional perturbation (compressing the wealth distribution toward equality). The paper formalizes these two positions as distinct mathematical optimization problems.

The analysis is restricted to the (C1)–(C3) class of tax schedules (Frøseth, 2026c), which preserves "generalized neutrality." Under these conditions, the tax system acts on the wealth process as a uniform drift shift (from the wealth tax) and a uniform rescaling of excess drifts (from corporate–dividend flow taxes). This restriction reduces the infinite-dimensional schedule space to a two-dimensional design plane parametrized by:

1. $k = (1 - \tau_c)(1 - \tau_d)$: The corporate–dividend retention factor (pass-through of flow taxes).
2. $\tau_w$: The proportional wealth-tax rate.

The objective is to identify the optimal point $(k, \tau_w)$ on a matched-revenue constraint set that minimizes two contrasting distortion criteria.

Methodology
The paper employs a continuous-time stochastic framework where pre-tax wealth follows a Geometric Brownian Motion (GBM). The post-tax log-wealth process is governed by a modified Fokker–Planck equation. The analysis compares two specific distortion metrics:

1. The JKO Free-Energy Gap ($\Delta F$): An information-theoretic measure defined as the difference in free energy between the post-tax and no-tax distributions at a finite horizon $T$. This aligns with the Mirrleesian tradition of minimizing decision distortion.
   $$\Delta F[\tau] := F[p^\tau_T] - F[p^0_T]$$
   where $F[p]$ is the sum of entropy and potential energy.

2. The Squared 2-Wasserstein Distance ($W_2^2$): A transport-geometric measure quantifying the "mass" required to transport the no-tax distribution to the post-tax distribution. This aligns with the Saez–Zucman tradition of distributional compression.
   $$W_2^2(p^\tau_T, p^0_T) = \int_0^1 [F^{-1}_{\tau}(u) - F^{-1}_{0}(u)]^2 du$$

The paper derives closed-form solutions for the minimization of both criteria subject to a linear revenue constraint $R \approx a\tau_w + b(1-k) = R^*$. It utilizes the properties of Gaussian distributions (preserved under the (C1)–(C3) class) to obtain explicit first-order conditions (FOCs).

Key Contributions and Results

1. Closed-Form Optima:
   The paper establishes that both minimization problems admit unique closed-form solutions within the (C1)–(C3) class.

   • The JKO optimum exhibits a structure that is linear in $\tau_w$ and involves a log-derivative in $k$.
   • The $W_2$ optimum is quadratic in both parameters.

2. The $\rho$-Crossover and Phase Structure:
   The behavior of the JKO optimum is governed by a single dimensionless ratio, the crossover ratio $\rho$:
   $$\rho := \Sigma_0 \cdot \frac{m_0}{\sigma^2} = \frac{\sqrt{v_0 + \sigma^2 T} (\mu - \sigma^2/2)}{\sigma^2}$$
   where $m_0$ is the geometric mean log-return, $\sigma$ is volatility, and $\Sigma_0$ is the horizon standard deviation.
   The JKO optimum partitions the regime axis into three distinct phases:

   • Pure Wealth-Tax Phase ($\rho < \rho_{low}$): The optimum sits at the corner $k=1$ (no flow tax) and maximal $\tau_w$.
   • Mixed-Instrument Phase ($\rho_{low} < \rho < \rho_{high}$): The optimum is an interior point utilizing both flow and wealth taxes.
   • Pure Flow-Tax Phase ($\rho > \rho_{high}$): The optimum sits at the corner $\tau_w=0$ and minimal $k$ (maximal flow tax).

   In contrast, the $W_2$ optimum is degenerate in this calibration: it pins to the pure flow-tax corner ($\tau_w = 0$) across the entire regime axis, exhibiting no phase transitions.

3. The Bluntness Mechanism:
   The paper identifies the economic mechanism driving the divergence between the two criteria as the bluntness index $B(m_0) = b/(a m_0)$, which measures the mean-displacement-per-revenue overshoot of the wealth-tax channel relative to the flow-tax channel.

   • The JKO criterion weights this bluntness linearly. It is willing to accept the wealth tax's displacement cost if the geometric return $m_0$ is sufficiently low.
   • The $W_2$ criterion weights bluntness quadratically. The squared displacement penalty makes the wealth-tax channel prohibitively expensive when $m_0$ is small, forcing the solution to the flow-tax corner.

4. Calibration and Regime Sensitivity:
   Using a "Norwegian-flavoured" calibration (representative of equity-heavy portfolios with $\sigma \approx 0.30$), the paper locates the regime parameter $\rho \approx 0.23$. This places the economy in the mixed-instrument phase for the JKO criterion, recommending a combination of a ~9% reduction in flow-tax pass-through and a ~2.7% proportional wealth tax.
   However, the paper notes that for typical households with significant real-estate holdings (lower effective volatility), $\rho$ increases, pushing the JKO recommendation into the pure flow-tax phase. The mixed-phase recommendation is thus shown to be fragile, sensitive to small shifts in portfolio volatility and the planning horizon $T$.

Significance and Claims
The paper claims to provide a precise mathematical articulation of the normative split between the Mirrleesian (allocational neutrality) and Saez–Zucman (distributional compression) traditions. It does not adjudicate which tradition is superior; rather, it demonstrates that the choice of distortion criterion fundamentally alters the optimal policy structure:

• Phase Richness: The JKO criterion generates a rich phase diagram with regime-dependent transitions, whereas the $W_2$ criterion yields a degenerate, single-phase solution in this setting.
• Regime Dependence: The optimal mix of wealth tax versus flow tax is not a fixed policy but a function of the economic regime (specifically the ratio of drift to diffusion).
• Neutrality vs. Compression: Within the neutrality-preserving (C1)–(C3) class, the disagreement is about the proportional mix of instruments. The paper notes that outside this class (as shown in the companion "Wasserstein paper"), the $W_2$ criterion would abandon neutrality entirely in favor of a progressive schedule, whereas the JKO criterion's behavior on the full class remains an open question.

The paper explicitly states that it does not compare its proportional recommendations to actual bracket-based wealth tax schedules in Norway or other jurisdictions, as those lie outside the (C1)–(C3) class. Such a comparison is deferred to a companion paper utilizing piecewise-Gaussian extensions. The current work serves to establish the theoretical baseline and the mechanism of the "bluntness" penalty that drives the divergence between information-theoretic and transport-geometric optimality.